The theory of proportional weirs - ERNET

d· Inst Sd July-Aug. 1995.75. 355-372. J In ,an . .. . o Indian Institute of SClence

The theory of proportional weirs

K. KESHAVA MURTHY

Departmenl of Civil Engineering. Indian Inslilule of Science. Bangalore 560 012. India.

Received on Augusl 18. 1994 ; Revised on Augusl 24. 1994.

Abstracl

This paper reviews the work of Keshava Munhy and collaborators in the area of proportional weirs. It brings forlh the lacuna that existed in the theory of weirs and focusses on why there was need a for a generalized theory of ,,"'(irs. It oullines the theorem of slope discharge continuity developed, emphasizing the importance of a datum or reference plane for every weir. It explains briefly the mathematical theory developed and its application to the design of several imponant weirs having wide application. The theory and application of geometrically simple weirs is briefly outlined.

Ke~"WOrds: Weirs. notches. proportional weirs. flow measurement. hydrometry. linear weir. quadratic weir. geo. metrically simple weir. .

1. Introduction

The study of weirs has been a subject of long-standing interest in hydraulics as is evidenced by the continuous flow of literature on this subject. The importance of weirs as a discharge-measuring device has been very well recognized . Sharp-crested weirs or nOlches are among the oldest, simplest and most accurate measuring devices used to measure the rate of flow in natural and artificial streams. It is well known that given any defined geometrical shape of a weir, the discharge through it can be found oul. This will be a function of h, the head causing flow. The conventional sharp-crested weirs of standard geometrical shapes like rectangular, triangular, trapezoidal and parabolic have been extensively investigated and their performance and characteristics well understood. But the reverse problem of finding the shape of a weir to produce a discharge which is a given function of h, called the 'problem of Ihe design of proportional weir', is of considerable interest in many fields like hydraulic, environmental and chemical engineering, and is, in general, not as simple and involves the solution of integral equations. The ~tudy of proportional weirs (P-weirs), besides having considerable practical application. ~s of fundamental and academic interest in hydraulics. The linear proportional weir with liS linear discharge head characteristics has been a subject of considerable interest. with lpphcations in diverse fields. Such weirs are used as control outlets for float-regulated ~Ing devices in chemical engineering, as a simple .measuring device in. hydraulics a~d , gallon and as an outlet for grit chambers rn envIronmental engrneenng to maIntaIn :on stant velocity in sedimentation tanks irrespective of fluctuations in discharge.

356 K. KESHAVA MURTHY

The first attempt to design a linear proportional weir was made by Oscar van p elt Stout in 1898 1 , while he was a professor of civil engineering at the University of N e- braska. He found out that the equation of such a weir is given by y oc xe 112,

where y and x are coordinates measured along the horizontal and vertical axes, respectively. This wei r, although theoretically exact, suffers from the practical difficulty of having an infinit e' crest width (y —4 co as x --> 0), which is physically unrealizable. Cowgill and Banks2.3 showed that the equation of the curve describing a weir producing a discharge Q = bir (m> 1/2), H being the depth of flow, is proportional to 171-312 . Stout's case can be obtained as a special case of this (in = 1).

In 1908, Sutro overcame the deficiency in Stout's model by providing a rectangular base of depth a and width 2W and fitting above this weir a designed complementary weir. For all flows through this weir (Fig. I) above the rectangular base, the discharge is proportional to the head measured above a 'reference plane' or 'datum' located at a13 above the weir crest. Though this worked out satisfactorily in this case, the rational basis for the selection of the datum was never explained. It was wrongly believed for over 50 years that the reference plane of the weir could be arbitrarily chosen. This erroneous notion was largely responsible for a lot of empiricism that crept into this important branch of hydraulics. The status of the subject was well summarized in 1966 by Singer and Lewis4 in the following words: "... In spite of its merits, the primary device is only known to a small number of specialists. There are several reasons for its relative obscurity, the most important one being the lack of up-to-date technical information. Techni- cal literature that exists is very old. Few textbooks on hydraulics have a chapter on proportional weirs, and the ones which do present the data in such a compressed form that one is rarely tempted to make further inquiries...." Though all these are true, the main reasons for its relative obscurity and empiricism are more complex. It is in this regard that the work on proportional weirs was taken up at the Department of Civil Engineering at the Indian Institute of Science in the late sixties to develop a theoretical understand- ing of the theory of proportional weirs. The aim of the project was two fold: (i) to develop mathematical theory of P-weirs, and (ii) to apply the same to design important weirs unsolved till then to prove its effectiveness.

As a part of the mathematical theory, the theorem on 'slope discharge continuity was recognized and proved. It states: "In any physically realizable weir having a finite number of finite discontinuities, the rate of charge of discharge is continuous at all points of discontinuity. The theorem has been proved rigorously using the theory of Laplace Transformsm and experimentally verified. Accordingly, every weir is associated with a unique reference plane or datum above which only are all heads reckoned. A new parameter X, called the 'datum constant', which fixes the datum is introduced. The choice of the datum at a13 above the crest in the case of Sutro weir is precisely meant to

satisfy the slope discharge continuity theorem although it was unrecognized.

1.1. Logarithmic weirs

The above mathematical theory was applied to design a logarithmic weir. From th works of Cowgill - and Banks 3 it is clear that any attempt to design a weir which has

ae

y-W[1- Fria. -1-44041 f ir

3.-2w3 5a it

Datum

s W

THE THEORY OF PROPORTIONAL WEIRS 357

FIG. I. Linear proportional weir (Sutro weir). FIG. 2. Rectangular-based quadratic weir.

term h" in the discharge equation Q = f(h), where trz < 3/2 invariably leads to a weir having infinite crest width. These are classified as 'compensating base weirs' requiring a base for their design. Logarithmic weir is one such weir. Using the general theory developed, the weir was designed with a rectangular base. Incidentally, logarithmic weirs give larger changes of head for a given change of flow, compared to conventional weirs and hence are useful in irrigation canals as sensitive measuring devices & 7 .

1.2. Quadratic weirs

The quadratic weir which gives a discharge proportional to the square root of the head- causing flow, has applications in bypass flow measurement, was first unsuccessfully tried by Haszpra 8 in 1965 in Hungary. An exact solution to the problem was given by Keshava Murthy9 . The function defining the weir has a very significant property of fast convergence leading a weir of zero width after a certain height, rendering it into a proportional orifice (Fig. 2). For all flows through this weir above the rectangular base of width nv and depth a, the discharge is proportional to the square root of the head measured above a datum located at 2a/3 above the crest, both while acting as a 'notch • as well

as when it is acting as an 'orifice'. A new concept of notch—orifice was introduced for

the first time. Several exact designs of quadratic weirs have been studied by Keshava Murthy et a1.6'9-11

1.3. Orifice-notch

!!1±"tigh the Sutro weir was used as an outlet weir for grit chambers or sedimentation nits_ to maintain constant average velocity necessary for the collection of the grit, it scu,attered from the main drawback in that it had to be fixed with its crest at the bed of the ttrl . risnel without leaving a clear gap of about 8-12 inches for the collection of the grit. "II was recognized as early as in 1936 12 and had remained unsolved.


a so that

This was taken up with the formulation and theory and a generalized synthesis p ro. cedure for the design of weirs having their base in any given shape to a depth the discharge through it is proportional to any singular monotonically increasing fine,.tion of the depth of flow measured above a certain datum. The problem is reduced t o finding out an exact solution of a Volterra integral equation in Abel's form. The maxi- mization of the datum below the crest of the notch was investigated. It was proved that for a weir notch made out of one continuous curve and for a flow proportional to the mth power of the head, it is impossible to bring the datum lower than (2m-1)a b elow the

crest of the notch.

A new concept of an 'orifice-notch' having discontinuity in the curve and having division of flow into two distinct portions was developed. The division of flow was shown to have a beneficial effect on the lowering of the datum below the crest and still main- taining the proportionality of flow s . This could be used effectively as grit chamber outlet weir, hence solving the long-standing problem in this regard.

1.4. Proportional v-notches

Although weirs for which the discharge Q cc , m 3/2, do not require a base (like conventional rectangular, v-notch, parabolic weirs), it was shown that they can also be designed with bases with advantage. These are classified as `noncompensating' base weirs'. Keshava Murthy and Pillai n ' 14 designed a modified proportional v-notch weir which produces the same head—discharge relationship, i.e., Q cc hs12 , as that of the conventional v-notch. They used a rectangular base which not only eliminates the difficulty of fixing the weir to plumb, but also increases the indication accuracy. Interestingly, it is seen that the proportional weir regains the geometrical simplicity of the conventional v-

notch weir, as the profile approaches fast a straight line. Other designs of noncompensating base weirs include proportional three-halves weir' s (to replace rectangular weir) and the proportional parabolic weir") .

2. Geometrically simple weirs

2.1. introduction

Linear proportional weirs have recently attracted considerable interest because of their

wide application in varied fields. Further, a linear proportional weir has greater indica -

tion accuracy in that a ± 1% error causes an equal percentage error in discharge compared to ± 1.5% error in rectangular weir and ± 2.5% error in v-notches. Although exact .

solution for the linear proportional weirs has been given by Sutro, Keshava MurtbY and, others 1.17-19 often these weirs are difficult to fabricate as they require sophisticate

d

equipment and skilled labour.

Recently, a few practical proportional linear weirs have been proposed bY Ramamurthym, Venkataraman and Subramanya 21 with the main objective of sin-Whir ing the weir geometry while incurring negligibly small errors in the discharge comPuta" tion.

THE THEORY OF PROPORTIONAL WEIRS a 359

f b , tcp limit Linearity

direfintY range • 94d 0.72 d

base ficw dePti .0.22±r2

Attitude d

Depth of flow h

_

1

f datum const4nt 1. a 0.0817 d

-CTial7rineEe 'lane or --T&itet;T:i— es ---- -

1

1t.3. Characteristics of the inverted v-notch as a linear proportional weir.

2.2. Inverted v-notch

The work of inverted v-notch was motivated by a casual remark by Troskolansky in his well-known book Hydrometry 22, in which he states that a closed trapezoidal weir with a vertex angle of approximately 50 0 has a near-linear head-discharge relationship. Surprisingly this had not been investigated. Neither its discharge—head relationship was known nor its range of validity. The inverted v-notch was analysed (Fig. 3) in the background of the general theory of proportional weirs. It was shown through an optimization procedure developed that the flow through this weir of half crest width W and depth d for depths above 0.22 d is proportional to the depth of flow measured from a reference plane situated at 0.08 d for all heads in the range 0.22 d hs0.94d within a maximum percentage deviation of + 1.5 from the theoretical dis- ch_ arge. Nearly 75% of inverted v-notch can be used effectively as the measuring range. Experiments are in very good agreement with the theory, giving a constant coefficient of discharge of 0.6223'24.

2.3. Chimney weir

111 e inverted v-notch was improved in the 'chimney weir' (Fig. 4) with respect to its r hange of applicability. It was shown that the range of linearity can be considerably enhanced by more than 200% by the addition of a rectangular weir of width 0.265 W (W is 7crest width of inverted v-notch) at a depth of 0.735 d (d is the altitude of the in- linear.v-notch)

above the crest of the weir 24, 25. The design parameters of the weirs, viz., jamm

ity range, base depth, reference plane, are estimated by solving the nonlinear pro- Using a numerical optimization procedure. It is shown that for flows

measnuraereshold depth of 0.22 d the discharges are proportional to the depth of flow the ran above a reference plane situated at 0.08 d above the weir crest for all heads in theoretiege,°-2.2 d h S 2.43 d, with a maximum percentage deviation of ± 1.5 from the disci

discharge. A significant result of the analysis is that the same linear head- kr relationship governing the flow through the inverted v-notch is also valid for the


2.43 d

Linearity range =2.21 d

Reference plane or Datum

0.735 d

\0:1117d

-441111--- W

Base depth at 0.22 d

FIG. 4. Salient features of a chimney plate weir.

extended chimney weir. Experiments with three different chimney weirs show excellent agreement with the theory, giving a constant coefficient of discharge for each weir.

The inverted v-notch weir and chimney weir are being tested in some minor irrigation canals in Karnataka.

2.4. Bell-mouth weirs

Troskolansky22 while referring to approximate weirs mentions that flow through the in -

tervening space obtained by keeping a semicircular disc in a rectangular channel Pi t"

duces an approximate linear head—discharge relationship. Unaware of this. Verikatacs man and subramanya2 have experimentally observed the linear discharge characterisn of weir called `quadrant plate weir'. However, no analytical investigation has bcefl


1

ii ........ t •

_. ..... lis- y

x A i i i .

i cui i

41- i .

_ _ I i x .

I I

1.

A °X z Head-h

WSW Profile X f(X)

„ma CI Fe-- we . 1•••••111•1

FIG.5. Typical EBM weir. FIG. 6. Definition sketch of the proposed weir.

made. A detailed theoretical analysis of the flow through the quadrant plate weir, referred to as bell-mouth weir (Fig. 5), is made in the light of the generalized theory of proportional weirs using the 'range of point method' of numerical optimization procedure. It is shown that the flow through the quadrant plate weir has a linear discharge— head relationship valid for certain ranges of head within a maximum deviation of ± 1% from the theoretical discharge. Further, it is shown through the optimization procedure that the measuring range of quadrant plate weir can be considerably enhanced by extend- ing the tangents to the quadrants at the terminals of the quadrant plate weirs. These are discussed in the paper on 'bell-mouth weir' by Keshava Murthy and Giridhar 24 ' 26 .

3. Self-basing linear weirs

3.1. Introduction

It. is clear from the works of Banks 3 and Cowgill 2 that the linear weir belongs to the :ass of 'compensating weirs' invariably requiring a base for its design, rendering it into !pes compound weir defined by two separate profiles. Further, it has been shown by k a..

heads muithy5.6 that these weirs have a unique reference plane or datum above which cads are reckoned. These weirs pass discharge, for all flows above the base weir,

pirlo

portional to the head above the datum.

nre?„!finiaattel there has been some special interest in the design and development of ap- 1"0:e_4ci e e linear weirs which produce near-linear head—discharge relationship i rtieturiefoirth referred to as h—q function). The prime motivation for this interest is geo- conditial simplicity and consequent ease in the fabrication, which is necessary in field

°Ilst where it is hard to find sophisticated equipment and skilled labour. Rama-


murthy et a/. 2° have designed a quadrant plate weir by replacing the curved profile of the Sutro weir' by the quadrant of a circle. Optimum dimensions of the weir have been found by minimizing the percentage deviation of discharge produced by the quadrant plate weir from the corresponding Sutro weir.

The first significant mention of a geometrically simple linear weir appears to be b y Troskolansky22 , where he refers to two approximate linear weirs: one an inward trape. zium (with an apex angle of 50°) and the second a quadrant weir (where the curved profiles are quadrants of a circle). It is said that these weirs can pass near-linear discharges in certain ranges of head. Keshava Murthy and Giridhar have analytically investigated in depth these two weirs, viz., the inverted v-notch 23 and bell-mouth weir 24.26 . In their study, they have used a numerical optimization procedure to fix the weir parameters governing the threshold depth (base depth) and the datum of the weir. The experiments fully confirm the results.

One of the main outcomes of the above investigation is the emergence of the linear weir defined by a single profile unlike the exact linear weirs like Sutro weir, etc. I.17-19 , which are essentially compound weirs defined by two profiles: one for the base and the other for the complementary weir above. It has to be underlined here that a portion of the profile of the weir above the crest itself acts as base for the weir. In other words, the base weir becomes an integral part of the whole weir itself. Hence, these weirs are ap- propriately called 'self-basing linear weirs'. One of the main drawbacks of the above designed self-basing weirs is that they have a limited range of head which makes their choice difficult in practice. The exact solution for the self-basing linear weirs is the one obtained by Stout', which, however, is physically unrealizable. In what follows, we are concerned with the finding of a very good approximate solution to this problem. Suc- cinctly, we seek an answer to the question 'Is it possible to find an approximate solution for a self-basing linear weir defined by one single profile with infinite measuring range?' It is shown that the significant property of rapid convergence (hence an approximate solution of Fredholm's integral equations of a particular kind) of a quadratic weir can be eminently exploited to arrive at a practical self-basing linear weir (henceforth referred to as SBL weirs) of very high degree of accuracy.

3.2. Preliminary consideration

Referring to Fig. 6, the discharge through the sharp-created weir (symmetrical about the x-axis), defined by y = f(x) for a head h is given by

( 1 ) q =2Cd irg-Sof(x). a—Cr dx,

where q is the discharge or rate of flow, h, the head above the weir crest, g, the accel- eration due to gravity, and Cd, the coefficients of discharge.

The coefficient of discharge is assumed to be constant (approximately equal to which is true for streamlined flows through sharp-crested weirs (this is confirmed by experiments).

later


Nondimensionalizing the above equation we have, for example,

Q = (H)ndX = 0(11), (2)

where

2Cd IfiWs5/2 '

Ws = half crest width,

ws

x=

For a proportional weir the hag function 0(H) is known a priori, in which case eqn (2) is the standard Volterra integral equation and can be reduced to Abel's form by differentiating with respect to H(using Leibnitz's rule), so that

f (X) dx = 20f(H)• Jo JH–X (3)

Solving eqn (3) 13, which is in Abel's form, we get the weir profile

f(X)=-2 lex (4)

The above solution is realizable only if 0(H) and OH) 0.C.X.c. co and 0(0) = 0' (0) = 0, where 0(H) is a continuous ing function of head.

are continuous in the range and monotonically increas-

33. Characteristics of the discharge function of the SRL weir

The discharge function of the SBL weir, in addition to satisfying the conditions specified iii the previous section, should satisfy another property, viz., it should tend to become linear very rapidly after a certain small threshold depth, the variation from linearity be- coining smaller as the head increases. In other words, the error in replacing this h–q relation (henceforth referred to as the primary haq function) by a linear one, should rapidly and continuously decrease as H increases. It has to be emphasized here that an x act linear relationship used as the primary h–q function results in a physically unreal-

izable form y cc x-I/2 (Stout profile', for which 1'(0) = °a).

3.4• Forms of primary h–q function

Theme.above-rnentioned near-linear property of the primary h–q function of the SBL weir 1-1 be realized

in three following forms (Fig. 7):

(a) S8L-1 (b) SEL-2 (C) SEL-3


FIG. 7. Forms of primary h—q function.

(i) A function although nonlinear in 0 X a rapidly becomes linear and retains this till infinity such that f(X) can be replaced by a form mX ± C in the range of a :c X < 03 with increasing accuracy, so that this function can be used directly as the primary h-q function 0(H) (Fig. 7a).

(ii) An increasing function which rapidly approaches its constant value at infinity such that this could be treated as the first differential of the primary h-q function, integrating which one can get the primary heti function 0(H) (Fig. 7b).

(iii) A continuously decreasing function which rapidly approaches zero, so that this could be treated as 0"(11) vs H curve, from which 0(H) can be obtained (Fig. 7c).

The above three forms of functions from which the primary h-q function of an SBL weir can be generated are termed for convenience as, SBL-1, SBL-2 and SBL-3-type generating functions, respectively.

3.5. Generating functions through exact solutions of some proportional weirs

In practice, it is very difficult to find A function which possesses all the essential proper- ties of a primary h-q function discussed above. However, some of the existing proportional weir functions (equations defining profile shape of the weirs) do posses certain special characteristics which can be exploited to generate the primary h-q function of SBL weir.

It has been observed that the profile of a proportional v-notch weir 13 designed using any base weir (Fig. 8a) is nearly a straight line after small values of x and the linearity rapidly improves with increasing x. Hence, by shifting the origin from 0 to 0' (Fig. 8a), the weir function considered with reference to the axes 0' X' and 0' Y (Fig. 8a) can be treated as SBL-1-type generating function

The quadratic weir9-11 , shown in Fig. 8b, has been found to become a 'proportiona l

orifice' or a weir of almost zero width for X > X0 (X0 is a small initial value), so that the function defining this weir considered with reference to the axes 0' X' (Fig. 8b) can be. treated as SBL-2-type generating function. Similarly, the function defining the quadratic


wei rOfi le with reference to the axes OX and OY (Fig. 8b) can be considered to repre- sent the S8L-3-type generating function.

3.6. Choice of generating function

An important aspect to be considered in the choice of a particular type of generating function is whether ether or not t results in a finite nonzero crest width for the designed linear weir. An infinite crest width is physically unrealizable and a zero crest width renders practical difficulties in the fixing of the weir to plumb and symmetry and in the accurate measurement of the initial head.

Cowgill 2 and Banks 3 have shown that an h—q relationship of the form Q oc Er can be obtained for a flow in a weir having its profile in the form of y cfc x' 312 , from which it is evident that the weir will have a finite crest width if and only if the least power of the head term in the discharge function 0(H) is 3/2.

It is found that out of the three types of the generating functions developed using the proportional v-notch and quadratic weirs, only for the SBL-2-type generating functions developed using the quadratic weirs the least power of head term is 3/2. For the rest it is greater than 3/2. In addition, the SBL-1-type generating function developed using the proportional v-notch is not amenable to exact integration and hence a closed-form solution of eqn (4) is not possible.

In the light of the above discussion, only SBL-2-type generating functions, developed using the exact solutions of quadratic weir, are considered for the design of the SBL weir.

4. Development of self-basing linear weir utilizing the exact solution of a quadratic weir

4.1. Generating the head discharge function

The function defining the profile of the parabolic-based quadratic weir" is (Fig. 9), for example,

2.17 y=1V 11-7-17- Va =fix). (12)

a Va x 1+ -

a

It has been mathematically proved 14 that f(x) is positive, single-valued and a continuously decreasing function. Further, the proportional weir transforms itself into a proportional orifice or a weir of almost zero width after a small finite height. Hence, by shifting the origin to 0' (Fig. 6)

. ..,

2 —

x' x ' lx— _ a ,

+ — + -- • a a x

1+• a . _.

(13) y i =


(a) Rectangular based proportional V-notch weir.

(b) Rectangular-based quadratic weir.

FIG. 8. Generating functions using profiles of P-weirs. (a) Rectangular-based proportional v-notch weir, (b) Rectangular-based quadratic weir.

• •

X I

Carpletneragy wet profile

_Y

Paratoic base

FIG. 9. Parabolic-based quadratic weir.

Nondimensionalizing, we get

x' • Y=1– 1-n+fiCet-

1+4X' — d (14) Y = W

an X =

Choosing the above function as SBL-2-type generating function to develop the self- basing weir, we have (Fig. 9),

O'(H) = 1 - + fir + 21ri (15) I + 411

Integrating eqn (1 5) with respect to H,

0(H) = H -32 1(1 + H) 312 - H312 + H1/21 _I tan el (21TO+ C. (16) 2

The constant of integration C is evaluated using the initial condition 0(0) = 0. It is found that C= 2/3. Therefore, the required primary h–q function is

„ 2 OW) = (n 2) - ! [(1 M312 ,H3/2 + H1/21 I

2 1 tan -1 (2a). ( 17)

Expanding the terms on the right-hand side of the above equation and simplifYing' we get

0(H) = 2H3/2 H 2 156 11 5/2 – 4 +.... 5 (18)


From eqn (18) it can be observed that the least power of the head term in the primary bi function is 3/2 and hence it is confirmed that the designed SBL weir will have a

finite, nonzero crest width.

4.2. To derive the function L (xr) defining the self-basing weir

Referring to Fig. 6, the discharge equation for flow through the proposed SBL weir in the nondimensional form is

10 H Jr—x- fs(x)dx = 0(H). (19)

Substituting for 0(H) from eqn (17),

iII 2 2 r ,,, i 1

Ins (X)dX = (H + —3

) — —3

i(1 + 10 312 — H 312 + Hate j— —2

tan -1 (2a). (20)

Differentiating with respect to H using Leibnitz's rule,

1111 is(X) Jo ndX=2[1-1-n+1 r 1+n— ' ° (H)* (21)

Equation (21) is recognized as the Volterra integral equation in Abel's form, whose solution is 13

fa) = —2 ix Ø"(H) dn. (22) z Jo frrii

Differentiating the RHS of eqn (21) and evaluating the integral in eqn (2) we get

fs (X) =1 — —2

tan -1 a- 2 + (23)

if (1+4X) 312 .

Equation (23) gives the profile of the SBL weir for flow through which the h—q relationship 0(H) is almost linear after a small base depth. The accuracy to which it is linear Can be best explained by comparing this theoretical h—q relationship with an exact linear h-q relationship.

4.3. Linear discharge characteristics of the designed weir It

is found that, beyond a small base depth, as the head increases, the h—q graph ap- proximates the asymptote (tangent at infinity) and the deviation of the value given by the tang at infinity from that given by the h—ci graph becomes increasingly negligible. Let

of dli.••• ■,.

(24) Qt. = mli ± C,

be the equation of the asymptote. Constants m and C and be found as .


FIG. 10. Parameters of the designed SBL weir.

. dQ . ] = 1 m= urn ein = 114 1+4H1- (25)

H-)** an H--+

and

22 1 C = lim (H - Q)= + -1(1+ H)312 H3/2 _ H 112 - - - 2

tan 1 (2a)] H-0 00 H-.00 3 3

fr C=---

2 =01187

' (26)

4 3

Hence, the equation of the asymptote, which is the linear relationship of the SBL weir. is

QL = H-0.1187. (27)

The absolute value of the deviation of this discharge from the one obtained by the theoretical h-q relationship given by eqn (17) is

Er(H)=Q-QL. (28)

It is seen that E(0) = finite, E(00) = 0 and

r

dQ = dildQ (29) dH = dH

0.05

0.02

0.01

Avg. CI a 0.6127 • -1% •

----------------------------------------------- _

0.615

0.610

0.605

• • • 10/ + 0


0.06

/ /

/ / .

/ /

/ I / /

/ / / A

I

f / h /

/

, .

. I

/ / T- i 40 cirn __—___...1 1 I 20 CM /

/ 1 /

-4E------ 120 an ---410- . . .

- ,

. i 4 i

..._ .

. . .

00 . /1 _ 0 1 1 1 , 1 1 1 1 _ 1 1 k 1 .1 1 1 l .- 1 1 • 11

--

head over ref. plane (m)

FIG. Ha. Discharge-head variation (experimental).

From eqn (14) it is found that the above expression is negative throughput the range OS I I C.3 • Hence, it is concluded that the relative error e given by ̀

e = Q QL (30)

continuously decreases as H increases in the range 0 < H 5 00, which is also evident from Fig. 5a. Further from Fig. 10a, it can also be observed that the error, which is nearly 1% at II re. 1 . 2, decreases very rapidly and is very nearly zero (accurate to third decimal place) at H = 4, beyond which for all practical purposes, the designed weir is as accurate as an exact linear weir. If a limit for the maximum permissible value of the error

0.620

0.1 0.2 0.4 0.6

0.1 0.2 0.3 head over ref. plane (m)

0.4 0.5 0.6

Flo. 1lb. Variation of C4 with head.


FIG. 12. SBL weir discharging.

(usually, ± 2%) is prefixed, the flow through the depth corresponding to any prefixed permissible maximum error (Fig. 10b) is analogous to the flow through the base weir of any existing exact linear weir. This depth is generally called the 'base flow depth' or the 'threshold depth (d)' of the designed SBL weir. For all flows above this threshold depth, the theoretical h—q relationship (eqn(17)) can be replaced by the linear relationship given by eqn (27). Hence, for all flows in the range d < h < 00, called the 'linearity range', the weir will pass discharges proportional to the head measured above a reference plane or datum located at 0.1187 above the weir crest (Fig. 10b). The validity of this linear relationship is further strengthened from the fact that the coefficient of discharge as obtained from experiments computed using the linear h—q relationship is constant to a high degree of accuracy.

4.4 Experiments

Experiments conducted on a typical SBL weir having a crest width of 40 cm in a rectan -

gular channel measuring 18.5 m long, 1.2 m wide and 1.1 m deep and with its crest 20 cm above the bed of the channel show excellent agreement with the theory, giv ing a constant average coefficient of discharge of 0.61 (Fig. 11). Figure 12 shows a phow° graph of the SBL weir discharging.


Acknowledg ements

me author is grateful to the authorities of the Indian Institute of Science, Bangalore, for necessary facilities for conducting this work, and to Prof R. Narasimha, FRS, roviding

for helpful suggestions. He thanks all his past and present students who have worked

P with him on this project.

References

I. Stour, 0. V. P.

2. COWGILL, A. P.

3. BANKS. R. B.

4. ShNGER, J. AND LEWIS, D. C. G.

5. KESHAVA MURTHY, K.

The proportional flow weir devised in 1896, Engng News Rec., 1914, 72, 148-149.

The mathematics of the weir forms, Q. Apple Math., 1944, 2, 142-147.

A note on generalised weir equation, Department of Civil Engineer- ing, North-Western University, paper HP 0654, Illinois, 1954.

Proportional-flow weirs for automatic sampling and dosing, Wat. Wat. Engng, 1966, 70(841), 105-111.

A generalised mathematical theory of proportional weirs, Ph. D. Thesis, Indian Institute of Science, Bangalore 560 012, India. 1967.

6. KESHAVA MURTHY K. AND

SESHAGIRI, N.

6. KFSHAVA MURTHY, K.

15. KESHAVA MURTHY, K. AND. JAYARAM, N.

A generalised theory and experimental verification of proportional notches, J. Franklin Inst., 1968, 285, 347-363.

- On the design of logarithmic weirs, J. Hydraul. Res., IAHR, 1966, 4. 51-59.

The problem of quadratic weir, proportional flow measurement, without disturbance of the water surface, Acta Tech. Acad. Sc!. Hung., 1965, 51, 121-132.

On the design of quadratic weirs, J. Franklin Inst., 1969, 287,

159-174.

Linear proportional weirs with trapezoidal bottoms, J. Hydraul.

Res., 1971, 9, 379-389.

Some aspects of quadratic weirs, J. Hydraul. Div., Proc. ASCE,

1977, 103, 1059-1076.

Sewerage and sewage treatment, p.421 1958, Asia Publishing House.

Quadratic weir having a parabolic base, Proc. 1X Aust. FM

Conf., Auckland, 1986, pp.554-556.

Modified proportional v-notch, J. Hydraul., Proc. Div., ASCE,

1978, 104,715-791.

Flow measurement with thin proportional weir, Ph.D. Thesis,

Bangalore 560 012, India, 1977.

plate and broad crested Indian Institute of Science.

A new investigation on parabolic weirs—propotional parabolic

weirs, Acta Tech., 1990, 103, 143-170.

7. GOVINDA RAO, N. S. AND

KESHAVA MURTHY, K.

8. HASZPRA, 0.

9. LAKSHAMANA RAO, N. S. AND

ABDUL BHUKARI, C. H.

10. KESHAVA MURTHY, K. AND

GOPALAKRISHNA PELLA', K.

11. 13A8Brr, H. E. AND BAUMANN, E. R.

13- KESHAVA MURTHY, K. AND DA!vIODARAN

NAMBOODIRIPAD, P. M


GOPALAKRISHNA PILLAI, K.

15.GOPALAKRISHNA PILLAI, K.

372

K. KESHAVA MURTHY

17. KESHAVA MURTHY, K.


GOPALAKR1SHNA PILLAI, K.

Proportional three-halves weir, J. Instn Engrs India, 1 988, 68, 231-234.

Design of constant accuracy linear proportional weir, J. Hydraut Div., Proc. ASCE, 1978, 104. 527-541.

19. SREENIVASULU, P. AND

RAGHAVENDRAN, R.

Linear proportional 1979, 96, 379-389.

weirs, J. Hydraul. Div., Proc. ASCE,

19. GIR1DHARA, D. P.

Quadrant-plate weirs, J. Hydraul. Div., Proc. ASCE, 1977, 103, 1431-1441.

A practical proportional weir, Wat. Power, 1973, 25, 189-190 .

Hydrometry: Theory and practice of hydraulic measurements, pp. 301-302, 1960, Pergamon Press.

Mathematical theory and experimental verification on geometrically simple and self basing linear weirs, Ph.D. Thesis, Indian Institute of Science. Bangalore 560 012, 1990.

Inverted v-notch: A practical proportional weir, J. Irrig. Drain. Engng Div., Proc. ASCE, 1989, 115, 1035-1050.

Improved inverted v-notch or chimney weir, J. &rig. Drain. En gig Div., Proc. ASCE, 1990, 116, 374-386.

20. RAMA MURTHY, A. S.,

SUBRAMANYA, K. AND PAN!, B. S.

21. VENKATARAMAN, P. AND

SURRAMANYA, K.

22. TROSKOLANSKI, A. T.


GIRIDHAR, D. P.


GIR1DHAR, D. P.


Geometrically simple linear weirs using circular quadrants: bell mouth GIRIDHAR, D. P. weirs, J. Hydraul. Res., IAHR, 1991, 29, 497-508.

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The theory of proportional weirs - ERNET